0.06e: Exercises - Rational Expressions
- Page ID
- 38228
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A: Simplify Rational Expressions.
Exercise \(\PageIndex{1}\)
\( \bigstar \) Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
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- Answers to odd exercises:
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1. \( 5 x ^ { 4 }; x \neq 0 \\[6pt]\)
3. \( \dfrac { x - 8 } { x + 8 }; x \neq -8 \\[6pt]\)
5. \( - \dfrac { 2 x + 3 } { x - 1 } ; x \neq 1, \dfrac { 3 } { 2 } \\[6pt]\)
7. \( \dfrac { 2 ( x - 7 ) } { 2 x - 1 }; x \neq -3, \dfrac { 1 } { 2 } \\[6pt]\)9. \( x - 1; x \neq -1 \\[6pt]\)
11. \(\dfrac { 11 } { 3 x ^ { 2 } ( 2 x - 5 ) } ; x \neq 0 , \dfrac { 5 } { 2 }\\[6pt]\)
13. \(\dfrac { x + 3 } { x - 7 } ; x \neq - 2,7\\[6pt]\)15. \(- \dfrac { x + 1 } { 5 x + 6 } ; x \neq - \dfrac { 6 } { 5 } , 1\\[6pt]\)
17. \(- \dfrac { 4 x + 3 } { x - 3 } ; x \neq \pm 3\\[6pt]\)
19. \(\dfrac { 1 } { x + 4 } ; x \neq 1 , \pm 4\\[6pt]\)
B: Multiply or Divide Rational Expressions.
Exercise \(\PageIndex{2}\)
\( \bigstar \) Multiply or divide as indicated, state the restrictions, and simplify.
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- Answers to odd exercises:
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21. \(\dfrac { 63 x } { x + 12 } ; x \neq - 12,0\\[6pt]\)
23. \(\dfrac { x - 8 } { 3 x ( x - 4 ) } ; x \neq - 8,0,4\\[6pt]\)
25. \(5 ( x + 1 ) ; x \neq - \dfrac { 5 } { 2 } , 0,2\)27. \(\dfrac { ( x + 7 ) ^ { 2 } } { 5 x ( x - 3 ) } ; x \neq - 3 , - 2,0,3\\[6pt]\)
29. \(- \dfrac { 5 x + 6 } { x - 3 } ; x \neq - \dfrac { 5 } { 4 } , - 1,1,3\\[6pt]\)
31. \(\dfrac { ( x + 6 ) ( 6 x - 1 ) } { ( x + 3 ) ( 2 x - 9 ) } ; x \neq - 3 , \dfrac { 1 } { 6 } , 2 , \dfrac { 9 } { 2 } , 5\)33. \(\dfrac { 25 } { a + b }\\[6pt]\)
35. \(\dfrac { 5 x \left( x ^ { 2 } - x y + y ^ { 2 } \right) } { x - y }\)
\( \bigstar \) Multiply or divide as indicated, state the restrictions, and simplify.
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- Answers to odd exercises:
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37. \(\dfrac { 5 x ( x + 3 ) } { x + 5 }\\[6pt]\) 39. \(\dfrac { 1 } { x + 5 }\\[6pt]\)
41. \(- \dfrac { 1 } { 2 x + 3 }\\[6pt]\)
C: Add or Subtract Rational Expressions.
Exercise \(\PageIndex{3}\)
\( \bigstar \) Add or subtract and simplify. State the restrictions.
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- Answers to odd exercises:
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51. \(\dfrac { 3 x + 2 } { 3 x + 4 } ; x \neq - \dfrac { 4 } { 3 }\\[6pt]\)
53. \(\dfrac { 1 } { x - 6 } ; x \neq - \dfrac { 1 } { 2 } , 6\\[6pt]\)
55. \(\dfrac { 1 - 2 x ^ { 2 } } { x } ; x \neq 0\\[6pt]\)57. \(\dfrac { -5 x+6 } { x - 1 } ; x \neq 1\\[6pt]\)
59. \(\dfrac { 2 ( x + 3 ) } { ( x - 2 ) ( 3 x + 4 ) } ; x \neq - \dfrac { 4 } { 3 } , 2\\[6pt]\)
61. \(\dfrac { ( x - 1 ) ( x + 2 ) } { x ^ { 2 } ( x - 2 ) } ; x \neq 0,2\)63. \(\dfrac { 2 x - 7 } { x ( x - 7 ) } ; x \neq 0,7\\[6pt]\)
65. \(\dfrac { x ^ { 2 } - 8 x - 5 } { ( x + 5 ) ( x - 5 ) ^ { 2 } } ; x \neq \pm 5\\[6pt]\)
67. \(\dfrac { x + 2 } { ( x + 4 ) ^ { 2 } } ; x \neq 0 , - 4\\[6pt]\)
\( \bigstar \) Add or subtract and simplify. State the restrictions.
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- Answers to odd exercises:
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69. \(\dfrac { 7 ( 5 - 2 x ) } { x ( 7 + x ) ( 7 - x ) } ; x \neq - 7,0,7\\[6pt]\)
71. \(\dfrac { x ( 5 x - 2 ) } { ( x - 4 ) ( x - 1 ) ( 2 x + 1 ) } ; \\ x \neq - \dfrac { 1 } { 2 } , 1,4\\[6pt]\)
73. \(\dfrac { x ^ { 2 } - 2 } { 2 x ^ { 2 } } ; x \neq 0\)75. \(\dfrac { x ^ { 2 } + 5 } { ( x + 2 ) ( x - 2 ) } ; x \neq \pm 2\\[6pt]\)
77. \(\dfrac { ( x - 1 ) ^ { 2 } } { x ^ { 2 } } ; x \neq 0\\[6pt]\)
79. \(\dfrac { 2 x - 1 } { x - 8 } ; x \neq - \dfrac { 1 } { 3 } , 8\\[6pt]\)81. \(\dfrac { x ^ { 2 } + 1 } { ( x - 1 ) ^ { 2 } ( x + 1 ) } ; x \neq \pm 1\\[6pt]\)
83. \(\dfrac { 2 x + 1 } { x } ; x \neq 0 , \dfrac { 1 } { 2 } , 1\\[6pt]\)
85. \(0 ; x \neq 0 , \dfrac { 2 } { 3 } , 2\)
\( \bigstar \) Add or subtract and simplify.
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- Answers to odd exercises:
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87. \(\dfrac { y ^ { 2 } + x ^ { 2 } } { x ^ { 2 } y ^ { 2 } }\\[6pt]\)
89. \(\dfrac { x + 2 y ^ { 2 } } { x y ^ { 2 } }\)91. \(\dfrac { x ^ { 2 } y ^ { 2 } + 16 } { x ^ { 2 } }\\[6pt]\)
93. \(\dfrac { 3 x ^ { 2 } + x + y } { x ^ { 2 } ( x + y ) }\)95. \(\dfrac { a + b - a ^ { 2 } } { a ^ { 2 } ( a + b ) }\\[6pt]\)
97. \(\dfrac { x ^ { n } + y ^ { n } } { x ^ { n } y ^ { n } }\)
D: Simplify Complex Rational Expressions.
Exercise \(\PageIndex{4}\)
\( \bigstar \) Simplify each complex rational expression.
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- Answers to odd exercises:
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101. \(\dfrac { 3 } { x ( x - 3 ) }\\[6pt]\)
103. \(\dfrac { x + 6 } { 8 x ^ { 2 } }\\[6pt]\)
105. \(\dfrac { 2 x - 5 } { ( x - 2 ) ( 5 x + 1 ) }\\[6pt]\)
107. \(\dfrac { x + 3 } { x - 5 }\\[6pt]\)109. \(\dfrac { 5 x ^ { 3 } } { x - 15 }\\[6pt]\)
111. \(\dfrac { 3 x } { x + 3 }\\[6pt]\)
113. \(- \dfrac { 6 y + 1 } { y }\\[6pt]\)115. \(\dfrac { x - 4 } { 3 x + 1 }\\[6pt]\)
117. \(\dfrac { 3 x - 2 } { 3 x + 2 }\\[6pt]\)
119. \(- \dfrac { 8 x - 1 } { x - 1 }\\[6pt]\)121. \(\dfrac { 3 x ( x - 3 ) } { ( x + 1 ) ( x - 1 ) }\\[6pt]\)
123. \(\dfrac { x } { ( x + 1 ) ( 3x - 1 ) }\\[6pt]\)
125. \(\dfrac { 4 x ^ { 2 } + 9 } { 12 x }\\[6pt]\)
127. \(\dfrac { 2 x - 5 } { 4 x }\\[6pt]\)
\( \bigstar \) Simplify each complex rational expression.
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- Answers to odd exercises:
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129. \(\dfrac { x + 1 } { 2 x + 1 }\\[6pt]\)
131. \(\dfrac { x y } { x + y }\\[6pt]\)133. \(- \dfrac { x + 5 y } { 5 x y }\\[6pt]\)
135. \(\dfrac { a ^ { 2 } b ^ { 2 } } { a ^ { 2 } - a b + b ^ { 2 } }\\[6pt]\)137. \(\dfrac { x y ( x + y ) } { x - y }\\[6pt]\)
139. \(\dfrac { x y } { x - y }\\[6pt]\)
141. \(\dfrac { 1 } { x + 1 }\\[6pt]\)143. \(\dfrac { x - 7 } { x - 5 }\\[6pt]\)
145. \(- \dfrac { 3 ( x - 2 ) } { 2 x + 3 }\\[6pt]\)
E: Mixed Practice with Rational Expressions.
Exercise \(\PageIndex{5}\)
\( \bigstar \) Perform the indicated operations. Simplify the result if possible. State the restrictions.
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- Answers to odd exercises:
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151. \(9 x ; x \neq 0\\[6pt]\)
153. \(- \dfrac { x + 8 } { 2 x + 1 } ; x \neq - \dfrac { 1 } { 2 } , 8\\[6pt]\)
155. \(\dfrac { x - 5 } { x } ; x \neq - 5,0 , \dfrac { 3 } { 2 }\\[6pt]\)
157. \(\dfrac { 4 x ( 2 x - 3 ) ^ { 2 } } { 2 x + 3 } ; x \neq \pm \dfrac { 3 } { 2 } , 0\\[6pt]\)159. \(\dfrac { 1 } { x + 6 } ; x \neq \pm 6\\[6pt]\)
161. \(\dfrac { 11 x - 5 } { 2 x ( x - 5 ) } ; x \neq 0,5\\[6pt]\)
163. \(- \dfrac { 1 } { 4 x - 5 } ; x \neq \dfrac { 5 } { 4 } , 3\\[6pt]\)
165. \(\dfrac { x - 5 } { x - 3 } ; x \neq 3,5\\[6pt]\)
167. \(\dfrac { t ^ { 2 } + 1 } { ( t + 1 ) ( t - 1 ) ^ { 2 } } ; t \neq \pm 1\\[6pt]\)
169. \(\dfrac { 2 x + 1 } { x ^ { 2 } } ; x \neq 0\\[6pt]\)
171. \(\dfrac { 7 x } { x - 7 }; \;x \ne 0, 7, -7\\[6pt]\)
173. \(\dfrac { ( x + 2 ) ( 2 x - 15 ) } { ( x - 5 ) ( 3 x - 4 ) }; \\ \;x \neq -2, 0, 5, \frac{4}{3}\\[6pt]\)
175. \(\dfrac { x ( x + 2 ) } { 2 x - 1 } \; d\neq 0, 2, \frac{1}{2} \\[6pt]\)
\(\star\)